# [seqfan] any value storing constants of power ratios?

Richard Mathar mathar at strw.leidenuniv.nl
Sun Feb 1 21:41:48 CET 2009

```Does it make sense to put infinite products of integer ratios involving
small integer powers into the OEIS? These may look like a too small sample
randomly chosen from a fuzzy set of other constants. The case of (n^2-1)/(n^2+1)
is already A090986, and (n^3-1)/(n^3+1)=0.6666666.. roughly equivalent to A020793,
therefore (n^4-1)/(n^4+1) is the smallest new value that comes into mind:

%I A000002
%S A000002 8,4,8,0,5,4,0,4,9,3,5,2,9,0,0,3,9,2,1,2,9,6,5,0,1,8,3,4,0,5,0,
%T A000002 0,7,7,0,5,8,4,7,9,8,7,4,8,6,8,8,4,7,1,7,6,6,6,4,3,0,6,9,6,4,5,
%U A000002 3,8,0,6,6,1,3,5,7,2,8,5,5,5,5,4,4,1,2,7,1,3,6,7,6,6,3,7,6,7,3
%N A000002 Decimal expansion of product_{n=2..infinity} (n^4-1)/(n^4+1).
%H A000002 E. W. Weisstein, <a href="http://mathworld.wolfram.com/InfiniteProduct.html">Infinite Product</a>, MathWorld.
%H A000002 J. Borwein et al, <a href="http://www.ams.org/mathscinet-getitem?mr=2051473">Experimentation in Mathematics</a>, 2004, section 1.2.
%e A000002 0.8480540493529003921296501834...
%O A000002 0,1
%Y A000002 Cf. A090986.
%K A000002 nonn,cons,base
%A A000002 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 01 2009

%I A000003
%S A000003 9,2,8,7,8,6,9,3,5,7,9,9,5,5,2,4,5,3,7,5,1,4,6,9,9,1,5,6,5,2,8,
%T A000003 5,2,3,5,1,9,3,2,0,1,0,1,5,0,3,7,5,3,0,4,1,1,8,2,0,1,0,2,8,2,6,
%U A000003 5,1,4,8,7,2,0,0,7,3,7,9,9,1,6,0,2,2,3,8,8,2,7,4,1,5,5,1,8,1,0
%N A000003 Decimal expansion of product_{n=2..infinity} (n^5-1)/(n^5+1).
%H A000003 E. W. Weisstein, <a href="http://mathworld.wolfram.com/InfiniteProduct.html">Infinite Product</a>, MathWorld.
%H A000003 J. Borwein et al, <a href="http://www.ams.org/mathscinet-getitem?mr=2051473">Experimentation in Mathematics</a>, 2004, section 1.2.
%e A000003 0.92878693579955245375146991...
%O A000003 0,1
%Y A000003 Cf. A090986.
%K A000003 nonn,cons,base
%A A000003 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 01 2009

%I A000004
%S A000004 9,6,5,9,0,6,0,8,5,7,6,2,1,5,9,2,1,2,1,5,7,0,6,2,3,7,0,5,5,0,4,
%T A000004 5,2,0,4,0,8,5,7,2,6,8,1,3,3,6,5,0,9,7,4,5,8,9,2,5,6,2,9,6,6,3,
%U A000004 9,4,9,2,7,3,7,7,0,2,4,9,0,0,7,5,7,3,0,9,3,2,7,1,1,1,0,7,7,7,1
%N A000004 Decimal expansion of product_{n=2..infinity} (n^6-1)/(n^6+1).
%H A000004 E. W. Weisstein, <a href="http://mathworld.wolfram.com/InfiniteProduct.html">Infinite Product</a>, MathWorld.
%H A000004 J. Borwein et al, <a href="http://www.ams.org/mathscinet-getitem?mr=2051473">Experimentation in Mathematics</a>, 2004, section 1.2.
%e A000004 0.96590608576215921215...
%O A000004 0,1
%Y A000004 Cf. A090986.
%K A000004 nonn,cons,base
%A A000004 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 01 2009

%I A000005
%S A000005 9,8,3,4,3,9,7,8,0,5,8,6,9,2,8,2,3,5,1,1,4,4,8,7,5,5,3,5,5,4,0,
%T A000005 1,3,6,3,5,3,7,3,1,5,4,1,6,2,8,6,8,8,3,1,1,2,1,9,0,3,7,6,0,4,5,
%U A000005 0,8,1,6,2,7,9,9,7,0,5,5,9,7,3,9,2,0,6,7,5,7,9,3,9,7,5,0,0,4,0
%N A000005 Decimal expansion of product_{n=2..infinity} (n^7-1)/(n^7+1).
%H A000005 E. W. Weisstein, <a href="http://mathworld.wolfram.com/InfiniteProduct.html">Infinite Product</a>, MathWorld.
%H A000005 J. Borwein et al, <a href="http://www.ams.org/mathscinet-getitem?mr=2051473">Experimentation in Mathematics</a>, 2004, section 1.2.
%e A000005 0.9834397805869282351144875535540...
%O A000005 0,1
%Y A000005 Cf. A090986.
%K A000005 nonn,cons,base
%A A000005 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 01 2009

%I A000006
%S A000006 9,9,1,8,7,8,4,0,7,6,5,8,2,9,4,8,1,6,9,6,6,4,2,6,9,2,7,7,8,4,8,
%T A000006 8,9,3,4,0,4,8,0,4,0,6,2,9,2,0,6,6,3,5,4,2,2,8,7,4,5,0,5,8,3,0,
%U A000006 7,6,1,9,5,8,1,8,4,1,2,5,0,3,9,1,6,8,5,9,7,3,1,0,8,1,8,9,9,1,6
%N A000006 Decimal expansion of product_{n=2..infinity} (n^8-1)/(n^8+1).
%H A000006 E. W. Weisstein, <a href="http://mathworld.wolfram.com/InfiniteProduct.html">Infinite Product</a>, MathWorld.
%H A000006 J. Borwein et al, <a href="http://www.ams.org/mathscinet-getitem?mr=2051473">Experimentation in Mathematics</a>, 2004, section 1.2.
%e A000006 0.9918784076582948169664...
%O A000006 0,1
%Y A000006 Cf. A090986.
%K A000006 nonn,cons,base
%A A000006 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 01 2009

%I A000007
%S A000007 9,9,5,9,9,1,2,6,5,8,9,3,4,0,4,6,5,5,1,1,9,1,9,4,1,3,1,7,4,5,8,
%T A000007 2,2,9,3,0,8,5,6,6,2,2,6,6,6,2,5,0,3,5,5,0,4,9,7,5,3,4,3,9,9,7,
%U A000007 1,9,6,3,6,6,5,1,6,1,7,2,7,3,5,1,1,6,2,3,2,8,3,3,6,4,3,9,7,7,9
%N A000007 Decimal expansion of product_{n=2..infinity} (n^9-1)/(n^9+1).
%H A000007 E. W. Weisstein, <a href="http://mathworld.wolfram.com/InfiniteProduct.html">Infinite Product</a>, MathWorld.
%H A000007 J. Borwein et al, <a href="http://www.ams.org/mathscinet-getitem?mr=2051473">Experimentation in Mathematics</a>, 2004, section 1.2.
%e A000007 0.99599126589340465511919...
%O A000007 0,1
%Y A000007 Cf. A090986.
%K A000007 nonn,cons,base
%A A000007 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 01 2009

%I A000008
%S A000008 9,9,8,0,1,2,8,2,6,1,7,2,9,8,2,7,8,4,1,9,0,0,3,9,8,1,4,5,0,8,9,
%T A000008 6,8,5,6,5,5,3,1,4,5,2,5,3,8,6,6,4,3,8,9,8,4,3,3,4,7,6,2,9,4,0,
%U A000008 3,4,9,5,1,1,7,1,7,2,8,6,1,2,5,7,0,6,6,4,6,6,2,2,7,4,4,2,6,4,4
%N A000008 Decimal expansion of product_{n=2..infinity} (n^10-1)/(n^10+1).
%H A000008 E. W. Weisstein, <a href="http://mathworld.wolfram.com/InfiniteProduct.html">Infinite Product</a>, MathWorld.
%H A000008 J. Borwein et al, <a href="http://www.ams.org/mathscinet-getitem?mr=2051473">Experimentation in Mathematics</a>, 2004, section 1.2.
%e A000008 0.99801282617298278419003981450896856553...
%O A000008 0,1
%Y A000008 Cf. A090986.
%K A000008 nonn,cons,base
%A A000008 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 01 2009

Richard J. Mathar, www.strw.leidenuniv.nl/~mathar

Digits := 120 :
m := 1:
for r from 2 to 10 do
omega := cos(Pi/r)+I*sin(Pi/r) :
x := (-1)^(m+1)*2*m*m!/r*mul( GAMMA(-m*omega^j)^(-(-1)^j),j=1..2*r-1) ;
x := Re(evalf(x)) ;
print(r,x) ;
od:

```